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My proudest moment in high school was getting a 5/5 on the calculus based AP Physics C exams at 15 with no calculus and only rudimentary algebra knowledge at th

My proudest moment in high school was getting a 5/5 on the calculus based AP Physics C exams at 15 with no calculus and only rudimentary algebra knowledge at the time. That experience permanently colored my thinking, and made me much more open to practicing thorough visual imagination as a way to solve problems. I found that practice useful all the way through my EE degree's vector fields courses a decade later. I think that's the modern fundamental difficulty in Maxwell's reworked equations - the 4 we all know and love, not the 20 or so he originally published. To even begin to get a true intuition for them, you have to get really really good at visualizing idealized objects with flows running through surfaces, and (if you're lucky) symmetries that cancel each other out. You can't be afraid of imagining the infinitely small and the continuous to really get the most out of it, even if you "know" on some deeper level that the continuity of spacetime is a convenient approximation. 14 years later I am still grappling with the beauty of saying "yeah yeah, this area of interest is technically discrete, but let's pretend it's continuous and see what kinds of stuff falls out." If you have examples of things like this in other areas like mathematical finance, I'd love to hear about them.

3abiton 3y ago
Can't agree more, mental visualization is such an asset for any understanding as it relies on compressing information and forces oneself to digest the material.
mjburgess 3y ago
> continuity of spacetime is a convenient approximation I disagree, and there's no evidence for this. This is computer science leaking out; physics has no formulation of spacetime in discrete terms, and indeed, all of physics presumes continuity. In QM, the space of wavefns is infinite-dim continuous, and if wasnt, QM wouldnt be linear. Cognition is discrete, but the world is continuous.
- yard2010 3y ago
  But isn't the whole point of QM is that this assumption doesn't hold in some scale? I mean it's literally in the name. Care to explain? :)
  - omnicognate 3y ago
    No, position and time are typically continuous variables in quantum mechanics. You can have formulations in which they are discrete but they are not required and are relatively exotic. QM certainly doesn't say they must be discrete.
    - hnfong 3y ago
      I feel that a lot of people here are confusing the math and "reality". You're definitely correct about the math, i.e. the systems that we humans have invented to model reality. But I guess most of us don't really care about what mathematical model scientists like to use (especially not whether they're "exotic" or not), but rather what reality could be like. And the quantum properties of QM do seem to suggest that there's some sort of fundamental discreteness in reality. And it seems to run contrary to the resolute claims that reality must be continuous as if it were a proven fact. What I understand is that the math most commonly used by scientists is definitely continuous, but whatever we can measure seems to have some kind of planck limitation. So are we talking about empirical science or science-flavored theology here? Have we actually found empirical evidence or proven the continuousness of space/time?
      - omnicognate 3y ago
        I responded to a couple of people who claimed with great certainty that QM meant spacetime had to be discrete, when it says nothing of the sort. I haven't claimed we have proof that it is continuous and I doubt we ever will as that seems akin to proving a negative existential. Your penultimate paragraph suggests some confusion about ideas like Planck scale and quantisation. Firstly, there is nothing special about the Planck length itself. It's just a unit of length. Around that sort of scale, though, our current theories of physics happen to break down because both quantum and gravitational effects become significant. That doesn't imply spacetime is discrete (or preclude it being discrete) at that scale. It's just a realm that our current theories don't work in. Secondly, while describing aspects of nature that are quantised was a large part of why quantum mechanics was developed (and the source of its name), it in no sense says anything like "there's some sort of fundamental discreteness in reality". Quantum mechanics deals with both discrete and continuous observables in a single framework: functional analysis, essentially. The set of possible values for an observable is modelled as the spectrum of an operator, which can be either continuous or discrete. Which sort of observable is appropriate for a given physical theory is a choice made in constructing that theory. For things like charge and spin we use discrete (quantised) values because we have evidence that those things are quantised. For things like position we use continuous values and have no evidence that using discrete observables would better match nature. Space could in reality be either discrete or continuous, or not even exist in any form we'd recognise as "space" on those scales. Quantum mechanics doesn't give us any hints one way or another.
- lll-o-lll 3y ago
  > the world is continuous Is it though? Does it matter one way or the other? Do we think reality is the math in some way, or is the math a really darn good model of the reality?
  - greysphere 3y ago
    Imagine there was a grid for space. For simplicity consider a regular grid of size 1unit in one direction and 1unit in a perpendicular direction. If such a grid existed, using one unit of ?something? would move you 1 unit along the axes of the grid, but you'd need 2 units of ?something? to move root2 units 45deg to the grid. Any discrete grid of any shape or size or pattern would have something like this, some sort of preferred alignment, but as far as we can tell there no such preference. Physics in free space is rotationally invariant and thus not on a grid thus continuous.
    - chr1 3y ago
      You don't have to imagine an ordered grid. If grid unit is small enough (say plank length 1,6 10^-35) and the grid is chaotic, for the distances of ~ 10^-16 that we can measure, everything will look the same in all directions. This happens the same way in which steel demonstrates isotropic behavior although its microscopic structure is anisotropic. So there is no easy way to prove or disprove continuity of space.
      - mjburgess 3y ago
        The "underlying issue" often at stake in the debate is whether reality is a computer, since it would need to be discrete if so, and often whether a computer can be made to simulate it. However, what's missed here is that discrete is a necessary but not sufficient condition. Once you give any sort of plausible account of how reality could be discrete, as you've done here, you end up with non-computable aspects (eg., typically randomness). So the metagame is lost regardless: reality isnt a computer (/ no complete physical theories of reality are computable). Though the meta-meta-game around "simulation" is probably internally incoherent in itself -- whether reality is a computer or not would really have nothing to do with whether any properties had by it (eg., mass) are simulated. Since either you take reality to have this property and hence "simulation" doesn't make sense, or you take it to be faked. If it's faked, being computable or not is irrelevant. There's an infinite number of conceivable ways that, globally, all properties could be faked (eg., by a demon that is dreaming).
- rsecora 3y ago
  Its 99 years since Einstein published the paper on the photoelectric effect whith had far-reaching consequences. [1] And 93 years since the first Solvay Conference. [2] [1] https://en.wikipedia.org/wiki/History_of_quantum_mechanics https://en.wikipedia.org/wiki/History_of_quantum_mechanics [2] https://en.wikipedia.org/wiki/Solvay_Conference https://en.wikipedia.org/wiki/Solvay_Conference
  - omnicognate 3y ago
    What's your point? Everything they did, including Einstein's (and everyone else's at the time) quantum mechanics work, was based on continuous space and time variables.
  - eigenket 3y ago
    Quantum mechanics does not mean everything is quantized. It got its name because the first predictions of quantum mechanics were quantized energy levels in some example systems, but that does not even mean that all energies are quantized in quantum mechanics. There are many systems you can study where energies are continuous, and many examples where other quantities are continuous in quantum mechanics.
- rini17 3y ago
  If it really was continuous so that physical quantities were real numbers as defined in mathematics, then it is in contradiction to maximal information density. Because almost all real numbers contain infinite amount of information. Full argument is elaborated here "Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real? by Nicolas Gisin": https://arxiv.org/abs/1803.06824 https://arxiv.org/abs/1803.06824
  - mjburgess 3y ago
    All the people who use thinks like the word "information" in this context are confusing thermodynamic, logical, probabilistic, (+ many others) and equivocating. "Information" is not a physical quantity, and there cant be a "volume" of it. Nor does this have anything to do with real numbers. It is impossible for there to be any system extended in space and time to "zoom infinitely" into a continuous range and hence record an infinite amount of information. No one claims this, and the formulation of physics (entirely on real numbers) does not require it. Rather to say, eg., space is continuous, is to say its unbroken. There is no physical quantity which is becoming infinite.
    - mikk14 3y ago
      Some people would disagree with dismissing information as non physical. For instance: https://scottaaronson.blog/?p=3327 https://scottaaronson.blog/?p=3327 The argument there would be that stuffing an extra bit of information in an information saturated volume would make it collapse into a black hole.
      - mjburgess 3y ago
        It's not entirely clear "Energy" is a physical property either. By physical I mean a causal property of a system which is a basic constituent of reality. For example in E = 1/2mv^2, a particle has kinetic energy in virtue of being matter in motion -- it is motion and matter which are basic. Energy is just a system of accounting which tracks motion in the aggregate over time (with kinetic/potential just being the future/past in the accounting) hence why energy conservation is just a temporal invarience. When making arguments about the physical properties reality has (eg., whether aspects are continuous) you need to be exceptionally clear what your terms mean, and terminology in physics isnt designed for this. There are no "information saturated volumes", this is a series of abstractions piled on top of each other. All the words in this area have quite complex formal definitions that are have quite difficult to unpack semantics, you cannot just go around saying "saturated volumes" -- it is this sort of language which breeds cranks, and pop sci does it with abandon. This entire discussion is a matter of several PhDs, and to be done only well by people with PhDs in the matter (philosophy of physics), or equivalent research. It's not possible to scrap fragaments of what compusci bloggers say and derive much that's likely to be actually correct.
- r0uv3n 3y ago
  > physics has no formulation of spacetime in discrete terms There are some attempts of working with discrete spacetime (e.g. causal set theory), but yeah, all our best descriptions so far very much assume smooth spacetime.
- omnicognate 3y ago
  > Cognition is discrete There's little evidence even of this, except in the trivial sense that language (minus prosody) is composed of discrete units.
  - mjburgess 3y ago
    It is discrete insofar as we're talking about sequences of thoughts, ie., reasoning. What offends the minds of some people is the world might not be like their mind at all. They want always to analogise everything to Reason. Everything should be countable, everything should be knowable, etc.
    - omnicognate 3y ago
      That's the "trivial sense" I'm talking about. If we restrict "cognition" to the stuff we know is discrete then trivially it's discrete. But cognition is a hell of a lot more than that.
      - lupire 3y ago
        I don't see how what we know is discrete. A word doesn't even have a discrete meaning, except locally in relation to other words. Saying A = B + C looks discrete, just by hiding any potential non-discreteness inside B and C.
        simiones 3y ago
        Discrete here means "non-continuous" - i.e. there is no smooth transition function between ideas/thoughts/rationalizations. For example, a formal proof is a discrete process: it follows step-wise rules that you can assign natural numbers to (this is the first step, this is the second step, this is the third step). A non-discrete process, a continuous one, would have a smooth transition between these steps, which is hard to even imagine. While I am not convinced it is correct to say that "human reasoning is discrete", human language is definitely discrete. Words don't blend smoothly into each other. If you don't believe me, try to define a function f:[0,1] -> Words, such that f(0) = "red" and f(1) = "blue" and tell me what is f(sqrt(2)/2), or what is df/dx.
- discoinverno 3y ago
  It is true that there is no experimental evidence, but I think there are some convincing arguments that something must happen at the Planck scale (for very short distances) in a full quantum-gravity theory. Here are some quotes from "Covariant Loop Quantum Gravity", Rovelli and Vidotto (slightly redacted). I suggest the whole chapter 1, in particular 1.2 to get an idea of why fundamentally spacetime may be discrete. "In general relativity, any form of energy E acts as a gravitational mass and distorts spacetime around itself. The distortion increases when energy is concentrated, to the point that a black hole forms when a mass M is concentrated in a sphere of radius R ∼ GM/c^2, where G is the Newton constant. If we take L arbitrary small, to get a sharper localization, the concentrated energy will grow to the point where R becomes larger than L. But in this case the region of size L that we wanted to mark will be hidden beyond a black hole horizon, and we lose localization. Therefore we can decrease L only up to a minimum value, which clearly is reached when the horizon radius reaches L, that is when R = L. Combining the relations above, [..] we find that it is not possible to localize anything with a precision better than the Planck length (~10^-35 m). Well above this length scale, we can treat spacetime as a smooth space. Below, it makes no sense to talk about distance. What happens at this scale is that the quantum fluctuations of the gravitational field, namely the metric, become wide, and spacetime can no longer be viewed as a smooth manifold: anything smaller than the Planck length is “hidden inside its own mini-black hole”." "The existence of a minimal length scale gives quantum gravity universal character, analogous to special relativity and quantum mechanics: Special relativity can be seen as the discovery of the existence of a maximal local physical velocity, the speed of light c. Quantum mechanics can be interpreted as the discovery [..] that a compact region of phase space contains only a finite number of distinguishable quantum states, and therefore there is a minimal amount of information in the state of a system. Quantum gravity yields the discovery that there is a minimal length lo at the Planck scale. This leads to a fundamental finiteness and discreteness of the world."
  - mjburgess 3y ago
    You're talking about minimum lengths, not discrete spacetime. It may be the case that there's a minimum length beyond which "no meaningful laws of physics apply", but it really says nothing about whether real numbers are indispensable in the formulation of physics, or about whether spacetime is continuous. There being a minimum length doesnt mean that everything is a discrete multiple of this length, or that space is broken into units of it, or that objects have to be aligned on grid boundaries defined by it. Whenever people try to do philosophy of physics the inevitable place everyone lands at is a series of false equivocations, often caused by the language of physics being ambiguous and polysemous. But "minimum length" here does not mean a sort of grid length.
- Keyframe 3y ago
  maybe we should take another look at analogue computing
- metricspaces 3y ago
  > computer science leaking out Planck constant would like to have a word with you. But it is true that CS shines a light on the matter of mapping the infinite into bounded spaces. This matter of ‘cognition’ is the entire matter (npi) of contention. What is the actual relationship between number and perceived phenomena? What is the deeper meaning of the concordance of mathematics and physics? Where do these magical constants come from and what does it all mean? It seems we bring the ‘world’ into being by partitioning. See Genesis 1 for details. > the world is continuous Reality is actually a unified undivided unity without form and timeless & eternal - that is all we can say with certainty. The “world” is our perception of this reality. Our cognitive machinary is discreet, and a mapping of this reality into metric & temporal spaces of the mind.
  - mr_mitm 3y ago
    > Planck constant would like to have a word with you. Do you want to flesh this out? Are you suggesting that because phase space is quantized, position space must be quantized as well?
    - metricspaces 3y ago
      A discreet unit of measure exists in dynamics & its relationship to position space is informed by Heisenberg’s theorem. (My actual point is that Reality is neither continuous nor discreet - it is an infinitesimal point and it is our mind — that likes to name and number things and relies on duality to make ‘distinctions’ — that creates the universe, the subjective reality that we perceive as inhabiting.)
      - lupire 3y ago
        What is an infinitesimal point?
        metricspaces 3y ago
        The tao that can be told is not the eternal Tao The name that can be named is not the eternal Name. The unnamable is the eternally real. Naming is the origin of all particular things.Free from desire, you realize the mystery. Caught in desire, you see only the manifestations. Yet mystery and manifestations arise from the same source. This source is called darkness. Darkness within darkness. The gateway to all understanding.
- lupire 3y ago
  Is wave function collapse continuous? Is photon absorption and emission continuous? No one knows what the universe of truly made of. Reality is measured up to certain error tolerances. Don't confuse the map (math and physics) with the territory (reality). Also, Stephen Wolfram would like a word with you.
  - mikhailfranco 3y ago
    Collapse is not continuous, by definition. Copenhagen is discontinuous. If you want continuity, then shun collapse, and believe in Many Worlds. You know you should. Wolfram is another conversation, and one that does not fit in this margin.
- TheOtherHobbes 3y ago
  This seems very unlikely. If you're from Copenhagen every measurement is a lossy discontinuity that resets the wavefunction. This is not an abstraction, it's directly observable. As for discrete formulations: https://en.wikipedia.org/wiki/Causal_dynamical_triangulation https://en.wikipedia.org/wiki/Causal_dynamical_triangulation
  - simiones 3y ago
    Interpretations of QM and measurement have very little to do with whether space-time is discrete or continuous. The simple fact is that no common QM formulation uses discrete mathematics for space-time, and it's unclear if any that does would even work. Also, your link is not a formulation of QM, it is a different theory which makes different predictions (it is a quantum gravity theory). And, per the sounds of the Wikipedia article at least, it is not actually proven equivalent to QM in the regimes where it needs to be ("There is evidence [1] that, at large scales, CDT approximates the familiar 4-dimensional spacetime", or in other words, it is not fully worked out if this is the case).
adrian_b 3y ago
The originally published equations were "20 or so" because one equation was written for each scalar component. Rewriting the equations in vector form reduces the number to the modern number. Moreover, the original equations are the complete system. The variant with 4 equations is the simplified variant for vacuum, which is mostly useless, except for the purpose of studying the propagation of electromagnetic radiation in vacuum. The complete system of equations has around 6 or 7 or even more equations, depending on whether one chooses to have distinct notations for various physical quantities, such as electric polarization, magnetization, current of free electricity carriers etc., or not. The variants with less equations are simplifications that are valid only in linear media, because only there you have proportionality relationships between quantities like electric field and electric polarization. Instead of learning a large number of simplified variants of the Maxwell equations with limited applicability, it would have been much better if a manual would present since the beginning the only complete variant that is always true, which must be in integral form, as initially published by Maxwell. The many simplified variants, for media without discontinuities where differential forms are valid, for stationary media, for linear media, for vacuum and so on, can be easily derived from the general form, while the reverse is not true.
- toth 3y ago
  > The originally published equations were "20 or so" because one equation was written for each scalar component. > Rewriting the equations in vector form reduces the number to the modern number. And if you use the differential form or 4d tensor notation they get reduced to 1 equation. Of course, for a lot of practical problems this is not very useful and it's better to work with the 3d vector form. > The variant with 4 equations is the simplified variant for vacuum, which is mostly useless, except for the purpose of studying the propagation of electromagnetic radiation in vacuum. > Instead of learning a large number of simplified variants of the Maxwell equations with limited applicability, it would have been much better if a manual would present since the beginning the only complete variant that is always true, which must be in integral form, as initially published by Maxwell. Here I have to strongly disagree. The version of Maxwell's equations that is fundamental and exactly correct [1] is the vacuum version. The ones with magnetization and displacement vectors are only approximations where you assume continuous materials that respond to fields in simple way. In truth, materials are made of atoms and are mostly vacuum: there is no actual displacement vector if you look close enough. Also the vacuum Maxwell equations are useful in many scenarios. For instance, that's how you compute the energy levels of Hydrogen atom or how you derive QED. Also, you have to start from them to derive the macroscopic versions with magnetization and displacement that you seem to like. [1] Well, up to non-linear quantum mechanic effects.
farseer 3y ago
I agree however as a commentator pointed out in a similar thread a few years ago, mental visualization only works for relatively small problems with limited variables. Most reasoning after that is done via equations when the problem complexity surpasses the n'th dimension (where n is maybe 3 or 4).
tim333 3y ago
Mathematical finance is basically all people saying look I know in the real world markets are driven by things like idiots buying Dogecoin because number go up but let's assume it's made of well informed participants who price everything correctly. Assuming this we can show...
lupire 3y ago
You may have taken a slightly wrong lesson from that exam. AP exams are extremely curved (scoring about 70% or less is a 5), and are half multiple-choice, so a few clues can get the answer), and it's partly a conceptual test that doesn't rely on math. But of course you're absolutely correct that continuous and discrete systems are approximately equal.
- blharr 3y ago
  AP Physics C is also one of the more heavily curved exams, where a 55-60% is a 5.
PaulHoule 3y ago
In quantum electrodynamics there is this problem: if you imagine an electron is a little sphere like the ball on a Van de Graff generator, there is a certain amount of energy in the electric field around it. As the radius gets tiny the field in the space immediately around it gets stronger so if you integrate it the energy of the EM field becomes infinite as the radius goes to zero…. We’ve got no evidence that the electron is more than a point, however. We use a trick called renormalization which, in this case, is recognizing that the mass of the electron has a term from the EM field. We’d assume that the EM theory is not completely true but that below some distance the theory breaks down. Working in momentum space there is a certain momentum that corresponds to the cutoff distance so we just don’t integrate beyond that. You can vary the cutoff and also vary the other parameters of the theory (such as the bare mass of the electron) so the theory gives the same answers at macroscopic distances so it doesn’t matter where you put the cutoff. Thus it does not matter much what the “true” theory is whether space is discrete or the electron really is a little ball or the EM field merges with the other forces at high energy to make some different force that (slowly) eats protons or quantum gravity or whatever. Discretization is problematic in a relativistic world because it breaks Lorenz invariance. That is, if I am moving quickly I would see the gap between the “pixels” get smaller. Now maybe the pixels can be non-Lorenz invariant but can “fake it” at low energies and large sizes but when the energy gets large you’d expect to see some evidence of the grain. Even if the gap was the Planck length you’d probably see things get weird at much lower energies, such as those of the highest energy cosmic rays. There has been a lot of research on that and there is no clear evidence of relativity being broken but it is still highly mysterious https://en.wikipedia.org/wiki/Greisen%E2%80%93Zatsepin%E2%80%93Kuzmin_limit https://en.wikipedia.org/wiki/Greisen%E2%80%93Zatsepin%E2%80... for instance Lorenz violation might allow particles to bypass that GZK limit.
1980phipsi 3y ago
I'm jealous. Unfortunately, people like me with aphantasia have no visual imagination. The hardest part in physics was converting the problems into equations. Once it is an equation, I could solve it (depending on the problem, with some difficulty or not, I guess).
seanhunter 3y ago
> If you have examples of things like this in other areas like mathematical finance, I'd love to hear about them There are tons of examples in mathematical finance but the obvious one is the Black/Scholes [1] paper where one of the key assumptions they make (which they know to be untrue but helpful) is that you can replicate a portfolio in continuous time. This allows them to use a constructed portfolio of a risk-free intrest-bearing instrument and the underlying to replicate the price of an option, and the process is a Brownian motion. Everyone knows that actual trading (and thus price processes) are discrete in real markets, but continuous time is much easier to model. Much later on people like Heston and Matytsyn(?sp) came up with stochastic vol models with jumps to replicate discrete price discontinuities, but they're a lot harder to work with in many ways. [1] https://www.cs.princeton.edu/courses/archive/fall09/cos323/papers/black_scholes73.pdf https://www.cs.princeton.edu/courses/archive/fall09/cos323/p...

3abiton 3y ago

Can't agree more, mental visualization is such an asset for any understanding as it relies on compressing information and forces oneself to digest the material.

mjburgess 3y ago

> continuity of spacetime is a convenient approximation I disagree, and there's no evidence for this. This is computer science leaking out; physics has no formulation of spacetime in discrete terms, and indeed, all of physics presumes continuity. In QM, the space of wavefns is infinite-dim continuous, and if wasnt, QM wouldnt be linear. Cognition is discrete, but the world is continuous.

yard2010 3y ago
But isn't the whole point of QM is that this assumption doesn't hold in some scale? I mean it's literally in the name. Care to explain? :)
- omnicognate 3y ago
  No, position and time are typically continuous variables in quantum mechanics. You can have formulations in which they are discrete but they are not required and are relatively exotic. QM certainly doesn't say they must be discrete.
  - hnfong 3y ago
    I feel that a lot of people here are confusing the math and "reality". You're definitely correct about the math, i.e. the systems that we humans have invented to model reality. But I guess most of us don't really care about what mathematical model scientists like to use (especially not whether they're "exotic" or not), but rather what reality could be like. And the quantum properties of QM do seem to suggest that there's some sort of fundamental discreteness in reality. And it seems to run contrary to the resolute claims that reality must be continuous as if it were a proven fact. What I understand is that the math most commonly used by scientists is definitely continuous, but whatever we can measure seems to have some kind of planck limitation. So are we talking about empirical science or science-flavored theology here? Have we actually found empirical evidence or proven the continuousness of space/time?
    - omnicognate 3y ago
      I responded to a couple of people who claimed with great certainty that QM meant spacetime had to be discrete, when it says nothing of the sort. I haven't claimed we have proof that it is continuous and I doubt we ever will as that seems akin to proving a negative existential. Your penultimate paragraph suggests some confusion about ideas like Planck scale and quantisation. Firstly, there is nothing special about the Planck length itself. It's just a unit of length. Around that sort of scale, though, our current theories of physics happen to break down because both quantum and gravitational effects become significant. That doesn't imply spacetime is discrete (or preclude it being discrete) at that scale. It's just a realm that our current theories don't work in. Secondly, while describing aspects of nature that are quantised was a large part of why quantum mechanics was developed (and the source of its name), it in no sense says anything like "there's some sort of fundamental discreteness in reality". Quantum mechanics deals with both discrete and continuous observables in a single framework: functional analysis, essentially. The set of possible values for an observable is modelled as the spectrum of an operator, which can be either continuous or discrete. Which sort of observable is appropriate for a given physical theory is a choice made in constructing that theory. For things like charge and spin we use discrete (quantised) values because we have evidence that those things are quantised. For things like position we use continuous values and have no evidence that using discrete observables would better match nature. Space could in reality be either discrete or continuous, or not even exist in any form we'd recognise as "space" on those scales. Quantum mechanics doesn't give us any hints one way or another.
lll-o-lll 3y ago
> the world is continuous Is it though? Does it matter one way or the other? Do we think reality is the math in some way, or is the math a really darn good model of the reality?
- greysphere 3y ago
  Imagine there was a grid for space. For simplicity consider a regular grid of size 1unit in one direction and 1unit in a perpendicular direction. If such a grid existed, using one unit of ?something? would move you 1 unit along the axes of the grid, but you'd need 2 units of ?something? to move root2 units 45deg to the grid. Any discrete grid of any shape or size or pattern would have something like this, some sort of preferred alignment, but as far as we can tell there no such preference. Physics in free space is rotationally invariant and thus not on a grid thus continuous.
  - chr1 3y ago
    You don't have to imagine an ordered grid. If grid unit is small enough (say plank length 1,6 10^-35) and the grid is chaotic, for the distances of ~ 10^-16 that we can measure, everything will look the same in all directions. This happens the same way in which steel demonstrates isotropic behavior although its microscopic structure is anisotropic. So there is no easy way to prove or disprove continuity of space.
    - mjburgess 3y ago
      The "underlying issue" often at stake in the debate is whether reality is a computer, since it would need to be discrete if so, and often whether a computer can be made to simulate it. However, what's missed here is that discrete is a necessary but not sufficient condition. Once you give any sort of plausible account of how reality could be discrete, as you've done here, you end up with non-computable aspects (eg., typically randomness). So the metagame is lost regardless: reality isnt a computer (/ no complete physical theories of reality are computable). Though the meta-meta-game around "simulation" is probably internally incoherent in itself -- whether reality is a computer or not would really have nothing to do with whether any properties had by it (eg., mass) are simulated. Since either you take reality to have this property and hence "simulation" doesn't make sense, or you take it to be faked. If it's faked, being computable or not is irrelevant. There's an infinite number of conceivable ways that, globally, all properties could be faked (eg., by a demon that is dreaming).
rsecora 3y ago
Its 99 years since Einstein published the paper on the photoelectric effect whith had far-reaching consequences. [1] And 93 years since the first Solvay Conference. [2] [1] https://en.wikipedia.org/wiki/History_of_quantum_mechanics https://en.wikipedia.org/wiki/History_of_quantum_mechanics [2] https://en.wikipedia.org/wiki/Solvay_Conference https://en.wikipedia.org/wiki/Solvay_Conference
- omnicognate 3y ago
  What's your point? Everything they did, including Einstein's (and everyone else's at the time) quantum mechanics work, was based on continuous space and time variables.
- eigenket 3y ago
  Quantum mechanics does not mean everything is quantized. It got its name because the first predictions of quantum mechanics were quantized energy levels in some example systems, but that does not even mean that all energies are quantized in quantum mechanics. There are many systems you can study where energies are continuous, and many examples where other quantities are continuous in quantum mechanics.
rini17 3y ago
If it really was continuous so that physical quantities were real numbers as defined in mathematics, then it is in contradiction to maximal information density. Because almost all real numbers contain infinite amount of information. Full argument is elaborated here "Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real? by Nicolas Gisin": https://arxiv.org/abs/1803.06824 https://arxiv.org/abs/1803.06824
- mjburgess 3y ago
  All the people who use thinks like the word "information" in this context are confusing thermodynamic, logical, probabilistic, (+ many others) and equivocating. "Information" is not a physical quantity, and there cant be a "volume" of it. Nor does this have anything to do with real numbers. It is impossible for there to be any system extended in space and time to "zoom infinitely" into a continuous range and hence record an infinite amount of information. No one claims this, and the formulation of physics (entirely on real numbers) does not require it. Rather to say, eg., space is continuous, is to say its unbroken. There is no physical quantity which is becoming infinite.
  - mikk14 3y ago
    Some people would disagree with dismissing information as non physical. For instance: https://scottaaronson.blog/?p=3327 https://scottaaronson.blog/?p=3327 The argument there would be that stuffing an extra bit of information in an information saturated volume would make it collapse into a black hole.
    - mjburgess 3y ago
      It's not entirely clear "Energy" is a physical property either. By physical I mean a causal property of a system which is a basic constituent of reality. For example in E = 1/2mv^2, a particle has kinetic energy in virtue of being matter in motion -- it is motion and matter which are basic. Energy is just a system of accounting which tracks motion in the aggregate over time (with kinetic/potential just being the future/past in the accounting) hence why energy conservation is just a temporal invarience. When making arguments about the physical properties reality has (eg., whether aspects are continuous) you need to be exceptionally clear what your terms mean, and terminology in physics isnt designed for this. There are no "information saturated volumes", this is a series of abstractions piled on top of each other. All the words in this area have quite complex formal definitions that are have quite difficult to unpack semantics, you cannot just go around saying "saturated volumes" -- it is this sort of language which breeds cranks, and pop sci does it with abandon. This entire discussion is a matter of several PhDs, and to be done only well by people with PhDs in the matter (philosophy of physics), or equivalent research. It's not possible to scrap fragaments of what compusci bloggers say and derive much that's likely to be actually correct.
r0uv3n 3y ago
> physics has no formulation of spacetime in discrete terms There are some attempts of working with discrete spacetime (e.g. causal set theory), but yeah, all our best descriptions so far very much assume smooth spacetime.
omnicognate 3y ago
> Cognition is discrete There's little evidence even of this, except in the trivial sense that language (minus prosody) is composed of discrete units.
- mjburgess 3y ago
  It is discrete insofar as we're talking about sequences of thoughts, ie., reasoning. What offends the minds of some people is the world might not be like their mind at all. They want always to analogise everything to Reason. Everything should be countable, everything should be knowable, etc.
  - omnicognate 3y ago
    That's the "trivial sense" I'm talking about. If we restrict "cognition" to the stuff we know is discrete then trivially it's discrete. But cognition is a hell of a lot more than that.
    - lupire 3y ago
      I don't see how what we know is discrete. A word doesn't even have a discrete meaning, except locally in relation to other words. Saying A = B + C looks discrete, just by hiding any potential non-discreteness inside B and C.
      - simiones 3y ago
        Discrete here means "non-continuous" - i.e. there is no smooth transition function between ideas/thoughts/rationalizations. For example, a formal proof is a discrete process: it follows step-wise rules that you can assign natural numbers to (this is the first step, this is the second step, this is the third step). A non-discrete process, a continuous one, would have a smooth transition between these steps, which is hard to even imagine. While I am not convinced it is correct to say that "human reasoning is discrete", human language is definitely discrete. Words don't blend smoothly into each other. If you don't believe me, try to define a function f:[0,1] -> Words, such that f(0) = "red" and f(1) = "blue" and tell me what is f(sqrt(2)/2), or what is df/dx.
discoinverno 3y ago
It is true that there is no experimental evidence, but I think there are some convincing arguments that something must happen at the Planck scale (for very short distances) in a full quantum-gravity theory. Here are some quotes from "Covariant Loop Quantum Gravity", Rovelli and Vidotto (slightly redacted). I suggest the whole chapter 1, in particular 1.2 to get an idea of why fundamentally spacetime may be discrete. "In general relativity, any form of energy E acts as a gravitational mass and distorts spacetime around itself. The distortion increases when energy is concentrated, to the point that a black hole forms when a mass M is concentrated in a sphere of radius R ∼ GM/c^2, where G is the Newton constant. If we take L arbitrary small, to get a sharper localization, the concentrated energy will grow to the point where R becomes larger than L. But in this case the region of size L that we wanted to mark will be hidden beyond a black hole horizon, and we lose localization. Therefore we can decrease L only up to a minimum value, which clearly is reached when the horizon radius reaches L, that is when R = L. Combining the relations above, [..] we find that it is not possible to localize anything with a precision better than the Planck length (~10^-35 m). Well above this length scale, we can treat spacetime as a smooth space. Below, it makes no sense to talk about distance. What happens at this scale is that the quantum fluctuations of the gravitational field, namely the metric, become wide, and spacetime can no longer be viewed as a smooth manifold: anything smaller than the Planck length is “hidden inside its own mini-black hole”." "The existence of a minimal length scale gives quantum gravity universal character, analogous to special relativity and quantum mechanics: Special relativity can be seen as the discovery of the existence of a maximal local physical velocity, the speed of light c. Quantum mechanics can be interpreted as the discovery [..] that a compact region of phase space contains only a finite number of distinguishable quantum states, and therefore there is a minimal amount of information in the state of a system. Quantum gravity yields the discovery that there is a minimal length lo at the Planck scale. This leads to a fundamental finiteness and discreteness of the world."
- mjburgess 3y ago
  You're talking about minimum lengths, not discrete spacetime. It may be the case that there's a minimum length beyond which "no meaningful laws of physics apply", but it really says nothing about whether real numbers are indispensable in the formulation of physics, or about whether spacetime is continuous. There being a minimum length doesnt mean that everything is a discrete multiple of this length, or that space is broken into units of it, or that objects have to be aligned on grid boundaries defined by it. Whenever people try to do philosophy of physics the inevitable place everyone lands at is a series of false equivocations, often caused by the language of physics being ambiguous and polysemous. But "minimum length" here does not mean a sort of grid length.
Keyframe 3y ago
maybe we should take another look at analogue computing
metricspaces 3y ago
> computer science leaking out Planck constant would like to have a word with you. But it is true that CS shines a light on the matter of mapping the infinite into bounded spaces. This matter of ‘cognition’ is the entire matter (npi) of contention. What is the actual relationship between number and perceived phenomena? What is the deeper meaning of the concordance of mathematics and physics? Where do these magical constants come from and what does it all mean? It seems we bring the ‘world’ into being by partitioning. See Genesis 1 for details. > the world is continuous Reality is actually a unified undivided unity without form and timeless & eternal - that is all we can say with certainty. The “world” is our perception of this reality. Our cognitive machinary is discreet, and a mapping of this reality into metric & temporal spaces of the mind.
- mr_mitm 3y ago
  > Planck constant would like to have a word with you. Do you want to flesh this out? Are you suggesting that because phase space is quantized, position space must be quantized as well?
  - metricspaces 3y ago
    A discreet unit of measure exists in dynamics & its relationship to position space is informed by Heisenberg’s theorem. (My actual point is that Reality is neither continuous nor discreet - it is an infinitesimal point and it is our mind — that likes to name and number things and relies on duality to make ‘distinctions’ — that creates the universe, the subjective reality that we perceive as inhabiting.)
    - lupire 3y ago
      What is an infinitesimal point?
      - metricspaces 3y ago
        The tao that can be told is not the eternal Tao The name that can be named is not the eternal Name. The unnamable is the eternally real. Naming is the origin of all particular things.Free from desire, you realize the mystery. Caught in desire, you see only the manifestations. Yet mystery and manifestations arise from the same source. This source is called darkness. Darkness within darkness. The gateway to all understanding.
lupire 3y ago
Is wave function collapse continuous? Is photon absorption and emission continuous? No one knows what the universe of truly made of. Reality is measured up to certain error tolerances. Don't confuse the map (math and physics) with the territory (reality). Also, Stephen Wolfram would like a word with you.
- mikhailfranco 3y ago
  Collapse is not continuous, by definition. Copenhagen is discontinuous. If you want continuity, then shun collapse, and believe in Many Worlds. You know you should. Wolfram is another conversation, and one that does not fit in this margin.
TheOtherHobbes 3y ago
This seems very unlikely. If you're from Copenhagen every measurement is a lossy discontinuity that resets the wavefunction. This is not an abstraction, it's directly observable. As for discrete formulations: https://en.wikipedia.org/wiki/Causal_dynamical_triangulation https://en.wikipedia.org/wiki/Causal_dynamical_triangulation
- simiones 3y ago
  Interpretations of QM and measurement have very little to do with whether space-time is discrete or continuous. The simple fact is that no common QM formulation uses discrete mathematics for space-time, and it's unclear if any that does would even work. Also, your link is not a formulation of QM, it is a different theory which makes different predictions (it is a quantum gravity theory). And, per the sounds of the Wikipedia article at least, it is not actually proven equivalent to QM in the regimes where it needs to be ("There is evidence [1] that, at large scales, CDT approximates the familiar 4-dimensional spacetime", or in other words, it is not fully worked out if this is the case).

adrian_b 3y ago

The originally published equations were "20 or so" because one equation was written for each scalar component. Rewriting the equations in vector form reduces the number to the modern number. Moreover, the original equations are the complete system. The variant with 4 equations is the simplified variant for vacuum, which is mostly useless, except for the purpose of studying the propagation of electromagnetic radiation in vacuum. The complete system of equations has around 6 or 7 or even more equations, depending on whether one chooses to have distinct notations for various physical quantities, such as electric polarization, magnetization, current of free electricity carriers etc., or not. The variants with less equations are simplifications that are valid only in linear media, because only there you have proportionality relationships between quantities like electric field and electric polarization. Instead of learning a large number of simplified variants of the Maxwell equations with limited applicability, it would have been much better if a manual would present since the beginning the only complete variant that is always true, which must be in integral form, as initially published by Maxwell. The many simplified variants, for media without discontinuities where differential forms are valid, for stationary media, for linear media, for vacuum and so on, can be easily derived from the general form, while the reverse is not true.

toth 3y ago
> The originally published equations were "20 or so" because one equation was written for each scalar component. > Rewriting the equations in vector form reduces the number to the modern number. And if you use the differential form or 4d tensor notation they get reduced to 1 equation. Of course, for a lot of practical problems this is not very useful and it's better to work with the 3d vector form. > The variant with 4 equations is the simplified variant for vacuum, which is mostly useless, except for the purpose of studying the propagation of electromagnetic radiation in vacuum. > Instead of learning a large number of simplified variants of the Maxwell equations with limited applicability, it would have been much better if a manual would present since the beginning the only complete variant that is always true, which must be in integral form, as initially published by Maxwell. Here I have to strongly disagree. The version of Maxwell's equations that is fundamental and exactly correct [1] is the vacuum version. The ones with magnetization and displacement vectors are only approximations where you assume continuous materials that respond to fields in simple way. In truth, materials are made of atoms and are mostly vacuum: there is no actual displacement vector if you look close enough. Also the vacuum Maxwell equations are useful in many scenarios. For instance, that's how you compute the energy levels of Hydrogen atom or how you derive QED. Also, you have to start from them to derive the macroscopic versions with magnetization and displacement that you seem to like. [1] Well, up to non-linear quantum mechanic effects.

farseer 3y ago

I agree however as a commentator pointed out in a similar thread a few years ago, mental visualization only works for relatively small problems with limited variables. Most reasoning after that is done via equations when the problem complexity surpasses the n'th dimension (where n is maybe 3 or 4).

tim333 3y ago

Mathematical finance is basically all people saying look I know in the real world markets are driven by things like idiots buying Dogecoin because number go up but let's assume it's made of well informed participants who price everything correctly. Assuming this we can show...

lupire 3y ago

You may have taken a slightly wrong lesson from that exam. AP exams are extremely curved (scoring about 70% or less is a 5), and are half multiple-choice, so a few clues can get the answer), and it's partly a conceptual test that doesn't rely on math. But of course you're absolutely correct that continuous and discrete systems are approximately equal.

blharr 3y ago
AP Physics C is also one of the more heavily curved exams, where a 55-60% is a 5.

PaulHoule 3y ago

In quantum electrodynamics there is this problem: if you imagine an electron is a little sphere like the ball on a Van de Graff generator, there is a certain amount of energy in the electric field around it. As the radius gets tiny the field in the space immediately around it gets stronger so if you integrate it the energy of the EM field becomes infinite as the radius goes to zero…. We’ve got no evidence that the electron is more than a point, however. We use a trick called renormalization which, in this case, is recognizing that the mass of the electron has a term from the EM field. We’d assume that the EM theory is not completely true but that below some distance the theory breaks down. Working in momentum space there is a certain momentum that corresponds to the cutoff distance so we just don’t integrate beyond that. You can vary the cutoff and also vary the other parameters of the theory (such as the bare mass of the electron) so the theory gives the same answers at macroscopic distances so it doesn’t matter where you put the cutoff. Thus it does not matter much what the “true” theory is whether space is discrete or the electron really is a little ball or the EM field merges with the other forces at high energy to make some different force that (slowly) eats protons or quantum gravity or whatever. Discretization is problematic in a relativistic world because it breaks Lorenz invariance. That is, if I am moving quickly I would see the gap between the “pixels” get smaller. Now maybe the pixels can be non-Lorenz invariant but can “fake it” at low energies and large sizes but when the energy gets large you’d expect to see some evidence of the grain. Even if the gap was the Planck length you’d probably see things get weird at much lower energies, such as those of the highest energy cosmic rays. There has been a lot of research on that and there is no clear evidence of relativity being broken but it is still highly mysterious https://en.wikipedia.org/wiki/Greisen%E2%80%93Zatsepin%E2%80%93Kuzmin_limit https://en.wikipedia.org/wiki/Greisen%E2%80%93Zatsepin%E2%80... for instance Lorenz violation might allow particles to bypass that GZK limit.

1980phipsi 3y ago

I'm jealous. Unfortunately, people like me with aphantasia have no visual imagination. The hardest part in physics was converting the problems into equations. Once it is an equation, I could solve it (depending on the problem, with some difficulty or not, I guess).

seanhunter 3y ago

> If you have examples of things like this in other areas like mathematical finance, I'd love to hear about them There are tons of examples in mathematical finance but the obvious one is the Black/Scholes [1] paper where one of the key assumptions they make (which they know to be untrue but helpful) is that you can replicate a portfolio in continuous time. This allows them to use a constructed portfolio of a risk-free intrest-bearing instrument and the underlying to replicate the price of an option, and the process is a Brownian motion. Everyone knows that actual trading (and thus price processes) are discrete in real markets, but continuous time is much easier to model. Much later on people like Heston and Matytsyn(?sp) came up with stochastic vol models with jumps to replicate discrete price discontinuities, but they're a lot harder to work with in many ways. [1] https://www.cs.princeton.edu/courses/archive/fall09/cos323/papers/black_scholes73.pdf https://www.cs.princeton.edu/courses/archive/fall09/cos323/p...